Czichos H., Saito T., Smith L.E. (Eds.) Handbook of Metrology and Testing

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408 Part C Materials Properties Measurement

7.5 Fracture Mechanics

In materials that are components of large engineering

structures the presence of fabrication ﬂaws and their

development during operational service should be an-

ticipated. It is rational to assume that ﬂaws exist and

that countermeasures against the fracture and fatigue

that develop from them should be taken.

In the 1920s, A. A. Grifﬁth conducted a series of

experiments to determine the fracture strength of glass

components that contained small ﬂaws. From this ex-

periments, he concluded that for brittle materials the

stress required to cause failure decreased as the size

of the ﬂaw increased, even after correction is made

for the reduced cross section of good material. Extend-

ing these ﬁndings to other types of materials, it was

generally found that the strength of a structural compo-

nent that contains a crack or crack-like ﬂaw decreases

with increasing crack size and cannot be deduced

from the mechanical properties obtained in conven-

tional tensile tests (Sect. 7.3.1) or other mechanical tests

using smooth specimens. The combined mathematical–

experimental approach employed in the evaluation of

the strength of cracked components is called fracture

mechanics; it provides a metodology to determine the

amount of crack growth and the residual strength of

cracked components during operational service.

7.5.1 Fundamentals of Fracture Mechanics

The fundamental principle of fracture mechanics is that

the stress and strain ﬁeld ahead of a sharp crack can be

characterized in terms of a single parameter, such as the

stress intensity factor K , that is a function of the applied

stress, crack size and geometrical boundary conditions.

Crack growth or fracture initiation from a precrack must

be controlled by the stress or strain state at the crack–

tip region. This implies that the fracture behavior from

a crack can be described by the fracture mechanics pa-

Applied stress

Flaw size

Fracture mechanics

parameter

Fracture resistance

of material

Yield or tensile strength

of material

Fig. 7.72a,b Comparison of the frac-

ture mechanics approach

(b) to design

with the (a) traditional design ap-

proach

rameter, as an analogy with the plastic yielding behavior

of materials described in the form of σ

(equivalent

stress) =σ

(yield strength of the material). The crack–

tip stress–strain similarity anticipates to result in the

same type of fracture under the same environmental

condition. Once the fracture resistance of a material is

described in terms of its fracture mechanics parameter

in laboratory tests, one can deduce its fracture behavior

under different applied stresses, crack sizes and geo-

metrical boundary conditions without any discussion of

the mechanisms of fracture concerned. The fracture me-

chanics approach has three variables for the evaluation

of strength, as shown in Fig. 7.72.

Fracture mechanics can be subdivided into two gen-

eral categories, namely linear elastic and elastic–plastic.

Materials with relatively low fracture resistance fail

under small-scale yielding at the crack tip and can

be analyzed through linear-elastic fracture mechanics

(LEFM), in which the stress intensity factor, K, is based

on the elastic stress analysis. Metallic materials of-

ten show large-scale plastic yielding preceding crack

growth or fracture initiation from the crack. Elastic–

plastic fracture mechanics (EPFM) is applied for these

large-scale yielding conditions. The J integral and the

crack–tip opening displacement (CTOD) can be used as

alternative fracture mechanics parameters to K.

Linear Fracture Mechanics

Consider an elastic body of arbitrary conﬁguration

with an ideally sharp crack, subjected to arbitrary

external forces. There are three types of relative move-

ments of the two crack surfaces, as Fig. 7.73 illustrates.

The opening mode, mode I, is the most popular

displacement mode in practical components, and is

characterized by opening displacements in the direction

of the principal load. Mode II and mode iII correspond

to in-plane shear loading and out-of-plane shear load-

Part C 7.5

Mechanical Properties 7.5 Fracture Mechanics 409

ing, respectively. In any problem a cracked body can be

treated as one or a combination of these three modes.

The theory of elasticity derives closed-form expres-

sions for the stresses around the crack tip for each of

these modes of deformation, assuming isotropic linear

elastic material behavior [7.228–230]. For a polar coor-

dinate axis as shown in Fig. 7.74, the stress ﬁeld in any

cracked body is given by

(r,θ) =







(θ)

∞



n=0







ij,n

(θ) , (7.78)

where σ

is the stress tensor, k and A

are constants,

and f

and g

are functions of θ. The higher-order

terms depend on the geometrical boundary conditions.

The ﬁrst leading term approaches inﬁnity as r reaches 0,

but the other terms remain ﬁnite. Thus, the stress at the

crack–tip region has a singularity of order 1/

√

r, regard-

less of the conﬁguration of the cracked body. The stress

ﬁelds ahead of a crack tip, where r is sufﬁciently small

compared with the crack length a, can be rewritten as

, I =

√

2πr

ij,I

(θ) , (7.79a)

, II =

√

2πr

ij,II

(θ) , (7.79b)

, III =

III

√

2πr

ij,III

(θ) , (7.79c)

for modes I, II, and III, respectively. These ﬁeld equa-

tions show that the magnitude of the elastic stress ﬁeld

can be described by the single-term parameters K

, K

and K

III

. The stress intensity factor K, given a subscript

to denote the mode of crack deformation, is a func-

tion of the applied force, the crack shape and size,

and the structural conﬁguration. However, since mode I

cracking is often question in practical constructions, the

subscript I is usually abbreviated for the mode I.

The general form of the stress intensity factor is

given by

K = Fσ

√

πa , (7.80)

where F is a parameter that depends on the geome-

try of the specimen or component with a crack. For an

inﬁnite plate subjected to uniform tensile stress σ con-

taining a through-thickness crack of length 2a, the stress

intensity factor is

K =σ

√

πa . (7.81)

a) b) c)

Fig. 7.73a–c Three modes of crack deformation. (a) Mode I (open-

ing) (b) Mode II (in-plane shear) (c) Mode III (out-of-plane shear)

Fig. 7.74 Crack–tip stresses in polar coordinate

Equations (7.79aa–c) show the elastic stress ﬁeld in the

singularity zone ahead of the crack for an idealized

crack model. In real materials, the stresses at the crack

tip do not take an inﬁnite value because the crack–tip

radius must be ﬁnite, and the metallic material in this

region undergoes plastic deformation and leads to fur-

ther relaxation of the crack–tip stress. However, the size

of the plastic zone or the ambiguity zone that surrounds

the crack tip is so small compared with the singular-

ity zone (for small-scale yielding) that the stresses in

the surrounding zone still take the form (7.79a). Thus,

the stress intensity factor K is still a representative pa-

rameter for the stress ﬁeld at the crack tip, although the

crack–tip stresses are devoured in black box.

The small-scale yielding condition is given by the

ASTM [7.231]as

a ≥2.5





, (7.82)

which gives one of the limitations for linear elastic frac-

ture mechanics.

Elastic–Plastic Fracture Mechanics

Under the large-scale yielding condition, where (7.82)

is not satisﬁed, the stress intensity factor loses its

physical basis as a fracture mechanics parameter. An al-

Part C 7.5

410 Part C Materials Properties Measurement

Fig. 7.75 Contour integral around the crack tip

ternative parameter for a nonlinear elastic material is the

J integral. Rice [7.232] presented a path-independent

contour integral, which is called J, for a cracked body.

The J integral is given by

J =





W(ε

)dy −T

∂u

∂x



(7.83)

where W(ε

) is the strain energy density, T

are

the components of the traction vector, u

are the

components of the displacement vector, and ds is

a length increment along the contour Γ , as illustrated

in Fig. 7.75.

The value of the J integral is independent of the path

of integration and is equivalent to the energy-release

rate for a nonlinear elastic material. It can be shown

that the J integral characterizes the crack–tip stress ﬁeld

in a nonlinear elastic material (the Hutchinson–Rice–

Rosengren (HRR) stress ﬁeld) [7.233, 234]aswellas

the stress intensity factor in a linear elastic material.

Under the large-scale yielding condition, the size of the

HRR stress ﬁeld depends on the magnitude of the plastic

yielding scale and the plastic constraint of a speci-

men. Whether the HRR stress ﬁeld exists or not gives

a limitation on the J integral approach for large-scale

yielding [7.235].

Another parameter in elastic–plastic fracture me-

chanics is the crack–tip opening displacement (CTOD),

the plastic deformation blunted an initially sharp crack,

as illustrated in Fig. 7.76. Wells [7.236] proposed the

opening at the crack tip as a measure of fracture tough-

ness. The CTOD can be related to the stress intensity

factor and the J integral under the small-scale yielding

condition as given by

J = mδσ

E(1−ν

)

, (7.84)

Plastic zone

Initial crack

Blunted crack

Fig. 7.76 Crack–tip blunting and CTOD

where δ is the CTOD, m is a dimensionless constant

that depends on material properties and stress triaxiality,

and E and ν are the elastic modulus and Poisson’s ratio,

respectively.

Details of the J integral and CTOD are well

described in [7.235]. To evaluate the J integral accord-

ing to (7.83), numerical analysis with a constitutive

equation for the material (relationship between plastic

strain and stress) is needed. The CTOD has an am-

biguity in its deﬁnition for numerical analysis. Thus,

the evaluation of parameters during the application

of the elastic–plastic fracture mechanics for actual

structures is difﬁcult. For a deeply cracked specimen

subjected to bending load, simple experimental proce-

dures have been proposed to evaluate the J integral and

CTOD [7.235,237, 238].

7.5.2 Fracture Toughness

The fracture mechanics parameter uniquely stipulates

the stress and strain state at the crack tip provided that

the constraint in a direction parallel to the crack front is

the same. The constraint on deformation in the crack–

tip region is closely related to the triaxial stress state and

depends on the geometry of the specimen, the scale of

plastic yielding, and strain hardening of the material.

Linear elastic fracture mechanics describes unsta-

ble fracture as fracture occurs when the stress intensity

factor K reaches a critical value K

. The fracture

toughness, K

for mode I deformation under small-

scale yielding and plane–strain stress state (sufﬁcient

constraint), is designated K

[7.231]. The toughness

in terms of the J integral is designated by J

and

[7.237]. J

is the toughness for onset of slow crack

growth from a crack and J

is the toughness for un-

stable fast fracture. It should be noted that there is

no harmonization in terminology of K

and J

. The

Part C 7.5

Mechanical Properties 7.5 Fracture Mechanics 411

CTOD criterion is also applied to evaluate the fracture

toughness under large-scale yielding. Historically, the

cleavage fracture toughness of structural steels has often

been evaluated in terms of the critical CTOD δ

In the standards for toughness testing, several spec-

iﬁcations are prescribed so that the constraint effect

can be negligible. Single-parameter fracture mechanics

breaks down in the presence of excessive plasticity, or

for specimens with too shallow a notch, and fracture

toughness depends on the size and geometry of the test

specimen.

Testing

The standard test methods for determining plane–strain

fracture toughness (K

) of metallic materials are spec-

iﬁed in several standards [7.231, 238, 239]. In these

standards, ASTM E399-90 [7.231], which was origi-

nally published as E399-70, is widely used.

For the standard K

test, a single-edge notched

(SEN) bend specimen or a compact test (CT) spec-

imen is used; these standard geometries are shown

in Figs. 7.77 and 7.78, respectively. The standard

bend specimen has a rectangular cross section (B ×2B,

W =2B, where B is the specimen thickness and W is

the specimen width). Alternative a specimen geometry

with square cross section (B × B, W = B) can also be

used for the bend test. However, the size requirement for

Notch

Fatigue

precrack

S B

Fig. 7.77 Single-edge notched (SEN) bend specimen

Fatigue

precrack

Notch

1.2 W

Fig. 7.78 Compact test (CT) specimen

the plane–strain fracture toughness test, which will be

described later, should be satisﬁed. In the CT specimen,

W = 2B is recommended.

The fatigue precrack is introduced at the notch tip

by cyclic loading and the recommended value for the

total crack length (machined notch plus fatigue crack)

is 0.45–0.55W. The fatigue precracking load can af-

fect the fracture toughness results, so the cyclic loading

conditions are strictly limited. The stress ratio (min-

imum load to maximum load) of cyclic loading is

between −1and0.1. The maximum fatigue stress in-

tensity (K

max

) does not exceed 80% of the K

value of

the material. For the ﬁnal stage of fatigue crack exten-

sion, K

max

must not exceed 60% of the K

value, and

the ratio of K

max

to the Young’s modulus of the material

max

/E should not exceed 0.00032 m

1/2

. Usually, the

number of cycles for fatigue precracking is 10

–10

Normally, as the temperatures of the fatigue cracking

and fracture toughness tests are different, the effect of

temperature should be taken into account. When fatigue

cracking is conducted at temperature T

and the frac-

ture testing at a different temperature, T

, K

max

must

not exceed 0.6(σ

ys1

/σ

ys2

, where σ

ys1

and σ

ys2

are

the yield stress at the respective temperatures T

and T

It is recommended that the length of the fatigue crack be

larger than 1.3mmand2.5% of W for a straight-through

notch.

The single-edge notched specimen is loaded in

three-point bending with a support span S, which is nor-

mally equal to 4W,thatisS =4W. The bend testing

ﬁxture is designed to minimize friction effects by allow-

ing the support rollers to rotate and move apart slightly

when the specimen is loaded. The compact specimen is

loaded in tension through pins by allowing the specimen

to rotate during the test.

Fracture tests are conducted with a static load-

ing with a rate of increase of stress intensity of

0.55–2.75 MPa m

1/2

. The test temperature is controlled

within 3

◦

C[7.237] and the specimen should be kept at

the test temperature for 1.2min/mm of specimen thick-

ness before testing [7.238]. During testing, continuous

measurement of load versus crack mouth opening dis-

placement is required. The displacement is measured

by a double-cantilever displacement gage or a ring-type

displacement gage.

After fracture testing, the crack length is measured

at the following three positions: at the crack front, and

midway between the center of the crack front and the

end of the crack front on each surface of the speci-

men. The average of these three measurements is the

crack length and is used to calculate the stress intensity.

Part C 7.5

412 Part C Materials Properties Measurement

Load P

000

Type I Type II Type III

95%

Secant

95%

Secant

95%

Secant

Displacement V

max

–P

Fig. 7.79 Principal types of load–displacement records

The maximum difference between these measurements

is speciﬁed in the standards.

Based on the test record of load versus crack mouth

opening displacement, it is ﬁrst necessary to calculate

a conditional result K

, using which the size require-

ment for the plane–strain condition is judged. When

the size requirement is satisﬁed K

becomes K

. The

procedure is as follows

1. Draw the secant line OP

from the origin of the

record with the slope (P/V )

=0.95(P/V)

, where

(P/V)

is the slope of the tangent OA to the initial

linear part of the test record, as shown in Fig. 7.79.

The load P

is then deﬁned as follows: if the load

at every point on the record which precedes P

lower than P

,thenP

is P

(Fig. 7.79, type I); if,

however, there is a maximum load preceding P

that

exceeds it, then this maximum load is P

(Fig. 7.79,

types II and III).

2. Calculate the ratio P

max

, where P

max

is the

maximum load of the record. If this ratio does not

exceed 1.10, that is,

max

≤1.10 (7.85)

then, proceed to calculate K

For an SEN bend specimen,

K =

1.5





f (x) =

0.5

2(1 +2x)(1 −x

1.5

)



1.99−2.15x +6.08x

−6.83x

+2.07x



x =

. (7.86)

For a CT specimen,

K =



0.5







f (x) =

2 +x

(1 −x)

1.5



0.886+4.64x −13.32x

+14.72x

−5.60x



x =

. (7.87)

3. Calculate 2.5(K

/σ

)

, where σ

is the yield

stress of the material at the test temperature. If the

following is satisﬁed, then K

is equal K

B, a ≥ 2.5





. (7.88)

4. If the test results fails to meet the requirements

in (7.80)orin(7.81), or both, it is necessary to use

a lager specimen to determine K

, J

Testing

When the specimen size is inadequate to meet the re-

quirement given by (7.88), a valid K

value is not

obtained. The reason for this is that the plastic deforma-

tion in the specimen is too large to meet the plane–strain

condition. In such case, an alternative method is to

use elastic–plastic fracture mechanics. The J integral

is one such parameters, which is a line or surface

integral enclosing the crack front from one crack sur-

face to the other, and is used to characterize the local

stress–strain ﬁeld around the crack tip. The method

for determining fracture toughness at the fracture in-

stability based on the J integral is described in the

standards [7.237, 238, 240], of which ASTM E1820-

01 [7.237] covers the procedures and guidelines for the

determination of the fracture toughness using fracture

parameters such as K, J, CTOD (δ). The original J

and J–R test methods were developed in the original

ASTM E813-81 [7.241] and E1152-87 [7.242], respec-

tively. Theses test methods are now uniﬁed in E1820.

When fracture instability occurs before stable tear-

ing the fracture toughness, labeled J

, is obtained.

When the instability occurs after stable tearing the frac-

ture toughness is denoted by J

.InJ

and J

testing, the

standard specimen for K

testing is applicable. How-

ever, in order to calculate J, it is necessary to record

the curves of load versus load-line displacement. For

Part C 7.5

Mechanical Properties 7.5 Fracture Mechanics 413

Fatigue

precrack

Notch

1.2 W

Fig. 7.80 Compact test (CT) specimen for the J integral

test

θ/2

Fig. 7.81 Double-clip gage method for determining load-

line displacement of the SEN bend test

CT test, the load-line displacement is directly measured

by using the specimen shown in Fig. 7.80.FortheSEN

bend test, special care must be paid to indentation of

the specimen by the loading jig. One possible method

for measuring the load-line displacement is to use dou-

ble clip gages. The load-line displacement V

may be

calculated by (Fig. 7.81)[7.238]

W(V

−V

)

−z

. (7.89)

For fatigue precracking, the maximum load should be

less than

max

≤ P

. (7.90)

is given by

0.5Bb

(7.91)

for the SEN bend specimen. For the CT specimen,

0.4Bb

2W +a

, (7.92)

Load P

Original slope

Load-line displacement V

Area A

Fig. 7.82 Deﬁnition of area for the J calculation

where, b

= W −a

(initial ligament depth) and σ

the ﬂow stress given as the average of the yield stress,

and the tensile strength σ

uts

. The condition (7.90)

is equivalent to limiting the maximum stress intensity,

max

of fatigue precracking, according to

max

≤0.47σ

0.5

. (7.93)

From the fracture instability point, the critical load P

is determined. The J-integral value J

corresponding

to P

is calculated using

J = J

+ J

, (7.94)

where J

is the elastic component of J and J

is the

plastic component of J. At a point corresponding to V

and P on the specimen load versus load-line displace-

ment, J is calculated through

J =



+η

, (7.95)

where, η =2fortheSEN bend, and η =2+0.522b

for the CT method. E



=(1−ν

)/E,andA

is the area

showninFig.7.82.

After testing, along the front of the fatigue crack and

the front of the stable crack extension (if it exists), mea-

sure the size of the original crack and the ﬁnal physical

crack size at nine equally spaced points centered about

the specimen centerline and extending to 0.005W from

the specimen surface. The original crack size a

and

the physical crack size a

are calculated by averaging

the two near-surface measurements and the remaining

seven crack-length measurements. The crack extension,

Δa

is calculated as a

−a

If Δa

< 0.2mm+J

/2σ

, the fracture instability

point corresponds to instability before stable tearing.

Part C 7.5

414 Part C Materials Properties Measurement

Load P Load P

or P

0.95 BD

0C D 0C D

Clip gage displacement V

Pop-in

that may

NOT

be ignored

Pop-in

that may

be ignored

Previously

ignored

pop-in

Fig. 7.83a,b Procedure for evaluating the signiﬁcance of pop-in

For a ferritic steel with 250 MPa <σ

< 725 MPa and

250 MPa <σ

uts

< 725 MPa for the room temperature

strength of the material,

B, b

≥50

, (7.96)

and for other metallic materials,

B, b

≥100

, (7.97)

where J

becomes J

. It should be noted that, even if

these conditions are met, J

might depend on thickness.

If Δa

> 0.2mm+ J

/2σ

, the fracture instability

point corresponds to instability after stable tearing and

becomes J

. This is a characteristic of the material

and specimen geometry and size.

When the material is heterogeneous, pop-ins are

sometimes observed in the record. If the pop-in is at-

tributed to an arrested fracture instability in the plane of

the fatigue crack, the result is considered to be a charac-

teristic of the material tested. Pop-in can be assessed

by a speciﬁc change in compliance, and also post-

test examination of the fracture surfaces. Pop-ins are

only evaluated when the load rises with increasing dis-

placement after pop-in, otherwise it is considered to be

a fracture instability.

The following procedure is used to assess the signif-

icance of pop-ins when post-test examination indicates

that they are associated with arrested fracture insta-

bility in the plane of the fatigue precrack. As shown

in Fig. 7.83: (1) draw a line CB which is parallel to the

initial slope OA and passes through the pop-in point un-

der consideration. (2) Draw a line CF with a slope 5%

less than the line CB. (3) Mark the point G, correspond-

ing to the load and displacement at pop-in. (4) When

the point G is within the angle BCF, the pop-in is

judged to be insigniﬁcant (Fig. 7.83b). (5) When the

point G is outside the angle BCF, the pop-in is signiﬁ-

cant (Fig.7.83a). The judgment of (4) and (5) is whether

the arrested crack is more than 0.02 of the crack length

or not.

J–R Curve Testing

The J–R curve shows resistance of a material to stable

crack extension in terms of the J integral and consists

of a plot of J versus crack extension. The standard test

method of ASTM E1820-01 [7.237] is to use a single-

specimen technique with partial unloading using an

elastic-compliance method to determine the stable crack

extension. The maximum J integral capacity of a spec-

imen J

max

and the maximum crack extension capacity

for a specimen Δa

max

are given by the following

max

=min



bσ

Bσ



, (7.98)

Δa

max

=0.25b

. (7.99)

In J–R testing, SEN and CT specimens are used, usu-

ally with side groove. Here, the net thickness is denoted

by B

and the effective thickness is expressed by

= B −(B −B

)

/B. The J–R curve is constructed

by plotting J integral values versus the correspond-

ing crack extension values, as shown in Fig. 7.84. The

unload/reload sequences are spaced at load-line dis-

placement intervals of about 0.005W. The J integral

Part C 7.5

Mechanical Properties 7.5 Fracture Mechanics 415

J (kJ/m

)

600

500

400

300

200

100

0123456

Crack extension (mm)

max

Δa

max

Fig. 7.84 Typical J–R curve

values are calculated by



+ J

pl,i

, (7.100)

pl,i



pl, j−1

i−1

pl,i

− A

pl,i−1





1 −γ

i−1

−a

i−1



, (7.101)

pl,i

− A

pl,i−1



i−1



pl,i

−V

pl,i−1

(7.102)

where, for SEN specimens η

=2andγ

=1, and for

CT specimens η

=2.0 +0.522b

i−1

/W and γ

=1.0 +

0.76b

i−1

/W. The quantity A

pl,i

− A

pl,i−1

in (7.102)is

the increment of plastic area under the load versus

load-line displacement record between lines of constant

displacement at points i −1andi, as shown in Fig. 7.85.

Using an elastic-compliance technique on an SEN

specimen with crack mouth opening displacement the

crack length is given as

=1.000−3.950u

+2.982u

−3.214u

+51.52u

−113.0u

(7.103)

where

u =



WEC

S/4



0.5

(7.104)

Load P

Plastic load-line displacement V

L, pl

i-1

L, pl, i-1

L, pl, i

Fig. 7.85 Deﬁnition of area for the J calculation

For CT specimens, the crack length is determined using

the load-line displacement as

=1.000+4.063u +11.24u

−106.0u

+464.3u

−650.7u

, (7.105)

u =





0.5

. (7.106)

Testing

The property J

characterizes the toughness of a ma-

terial near the onset of crack extension from a fa-

tigue precrack. Originally, the J

test was speciﬁed

in ASTM E813 [7.241] and is now uniﬁed in ASTM

E1820 [7.237]. E1820-01 gives two methods to deter-

mine J

: the basic procedure and the resistance curve

procedure. The basic procedure involves physical mark-

ing of the crack extension and multiple (more than ﬁve)

specimens, which are used to develop a plot from which

an initiation toughness value is evaluated. The resis-

tance curve procedure is to determine the J

value from

the J–R curve, which is an elastic-compliance method

where multiple points are determined from a single spec-

imen.

In the basic procedure, ﬁve or more specimens are

used. After unloading the specimen, the crack is marked

by heat-tinting at about 300

◦

C (for steels and titanium

alloys) or fatigue cracking, and then, cooling and break-

ing the specimen. The physical crack size a

is then

measured at nine equally spaced points along the front

of the marked region of stable crack extension. The crack

extension Δa is calculated with

Δa =a

−a

. (7.107)

Part C 7.5

416 Part C Materials Properties Measurement

J (kJ/m

)

800

600

400

200

0 0.5 1.0 1.5 2.0 2.5

Crack extension (mm)

1.5 mm exclusion line

0.2 mm offset line

0.15 mm

exclusion line

Construction

line

Points used for

regression analysis

Power law regression line

limit

Mσ

Δa

min

Δa

min

Fig. 7.86 Deﬁnition of construction line for data qualiﬁcation

J (kJ/m

)

800

600

400

200

limit

Region of qualified data

0.15 mm

exclusion line

Construction

line

0.5 1.0 1.5 2.0 2.5

Crack extension (mm)

1.5 mm exclusion line

Δa

min

Δa

min

Fig. 7.87 Deﬁnition of regions for data qualiﬁcation

The J value is calculated using (7.95) at the point where

unloading was conducted.

In the resistance curve procedure, a correction is ap-

plied to the estimated Δa

data to obtain an improved

original crack size, a

from the compliance. Using the

data set of J

and a

, a

is calculated from

a =a

2σ

+BJ

+CJ

. (7.108)

The coefﬁcients B and C in this equation are deter-

mined using a least-square ﬁt procedure. In this proce-

dure, it is necessary that there are more than eight data

and that three of the eight are between 0.4J

and J

Moreover, the correlation coefﬁcient of this ﬁt should be

more than 0.96 and the difference between the optically

measured crack length, a

and a

should be less than

0.01W. For each a

value, the crack extension is calcu-

lated by

Δa

−a

. (7.109)

The J

values are determined from (7.100–7.102).

The next step is to plot J versus Δa,asshown

in Fig. 7.86. Draw a construction line in accordance with

J = Mσ

Δa , (7.110)

where M =2. Then, draw exclusion lines parallel to the

construction line intersecting the abscissa at 0.15 mm

and 1.5 mm, as shown in Fig. 7.87. These lines give the

minimum and maximum Δa values (Δa

min

, Δa

max

). At

least one J–Δa point should lie between the 0.15 mm

exclusion line and the 0.5 mm offset line, and one point

should also lie between the 0.5 mm offset line and the

1.5 mm exclusion line. Next we draw the regression line

with

ln(J) =ln(C

) +C

ln(Δa) . (7.111)

The J

value is determined as the intersection of the re-

gression line of (7.111)andthe0.2 mm offset line. If the

following size requirements are satisﬁed, J

becomes

B, b

> 25J

/σ

. (7.112)

CTOD Testing

The CTOD test is especially appropriate for materials

that exhibit a change from ductile to brittle behav-

ior with decreasing temperature. Typical standards of

CTOD testing method are speciﬁed in [7.238,243,244].

The method uses fatigue precracked specimens.

A rectangular- or square-cross-sectioned three-point

bend specimen or compact specimen is used. The con-

ﬁgurations of the specimens are similar to those for K

testing (Figs. 7.77 and 7.78). A fatigue precrack is intro-

duced half way along the specimen width, 0.45–0.55

times the specimen width. In order to expedite fatigue

precracking, a machined notch is produced by milling,

sawing or disc grinding. A chevron notch may be ma-

chined if fatigue precracking is difﬁcult to control. The

fatigue precrack is introduced by repeated loading at

room temperature. The maximum fatigue precracking

load should be lower than a prescribed value. This re-

striction is to limit the plastic zone size at the fatigue

Part C 7.5

Mechanical Properties 7.5 Fracture Mechanics 417

precrack tip. Excessive plastic deformation at the crack

tip may inﬂuence the critical CTOD value.

The specimens are tested in displacement-controlled

and quasistatic monotonic loading. The machine for

load application should be capable of applying load

at a rate of stress intensity factor within the range

0.5–3.0MPam

1/2

−1

. A typical set up of a specimen

is shown in Figs. 7.77 and 7.78.

The notch opening displacement is measured by

a displacement gage attached to the edge of the notch.

Knife-edges are attached at the notch edge. Alterna-

tively, integral knife-edges are used. A record of load

versus notch opening displacement is made. Character-

istic types of load versus displacement records in the

test are shown in Fig. 7.88. Note that P

is the applied

load at the onset of brittle crack extension or pop-in

when the average stable crack extension (Δa), including

the stretched zone width (SZW), is less than 0.2mm;

is the applied load at the onset of brittle crack ex-

tension or pop-in when the event is preceded by Δa

of at least 0.2mm; and P

is the applied load at the

ﬁrst attainment of a maximum load plateau for fully

plastic behavior. If the specimen fractures by brittle

crack extension prior to the ﬁrst attainment of a max-

imum load plateau (type (4)), the fracture surface is

examined for evidence of stable crack extension in

the region between the fatigue precrack front and the

start of brittle crack extension. When an initiated brittle

crack is arrested after a short extension, the load ver-

sus notch opening record exhibits pop-in, see types (3)

and (5) in Fig. 7.88. Pop-ins giving both a load drop,

y, and displacement increase, x, of less than 1% are

ignored.

The types and the amount of Δa are recorded. The

critical values of the load and notch opening displace-

ment of F

and V

,orF

and V

, are measured at points

corresponding to

1. Fracture, when there are no signiﬁcant pop-ins:

record types (1), (2) and (4);

2. The earliest signiﬁcant pop-in prior to fracture:

record types (3) and (5);

3. Fracture, when all signiﬁcant pop-ins prior to frac-

ture give values of d%F that are less than 5%, where

d%F =100



1 −



F −y

D +x



% . (7.113)

Val ue s o f P

and V

are measured at points corre-

sponding to

Load P

Notch opening displacement V

(1) (2) (3) (4) (5) (6)

Fracture

Fig. 7.88 Characteristic types of load–displacement records in

a CTOD testing (after [7.238])

Load P

, V

or V

, P

or P

Parallel

to 0A

Fig. 7.89 Deﬁnition of V

for determining the CTOD (af-

ter [7.238])

•

First attainment of the maximum load plateau, when

there is no fracture or pop-ins prior to the ﬁrst attain-

ment of the maximum load plateau: record type (6).

Critical CTOD values, δ

, δ

, corresponding to

and V

, P

and V

,orP

and V

, respectively, are

calculated by

δ =



1.5

× f







1 −ν



2σ

0.4



W −a



0.4W +0.6a

, (7.114)

where, P is the critical value of the applied load, S is

the bending span, B and W are the thickness and width

of the specimen, respectively (Fig. 7.77), a

is average

original crack length, f is a mathematical function of

/W (7.86)[7.238], σ

is 0.2% of the proof strength at

Part C 7.5